NEWCOMB’S PARADOX
For several years William Newcomb and I were workplace colleagues. I first learned of his famous paradox from him, and I now describe the paradox in the way that I remember him telling it to me:
• Suppose that there is a Being — whom you may think of as God if you wish — who knows how you think so completely and in such detail that he is able to foresee what you would do under any and all circumstances. This Being shows you two closed boxes. One box is transparent, and you can see that it contains $1,000. The other box is opaque, and the Being — who never lies or cheats — tells you that it contains either $1,000,000 or nothing and that he already has fixed its contents and will not change them. You are to choose between taking the contents of either (1) only the opaque box or (2) both boxes. The Being tells you that, if he foresaw that you will take from (2) both boxes, then he has put nothing into the opaque box; otherwise, if he foresaw that you will take from (1) only the opaque box, then he has put $1,000,000 into it. What should you do?
Philosopher Robert Nozick remarked, "To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly." On the one hand, either the opaque box already contains $1,000,000, or it does not; so taking from both boxes must yield $1,000 more than taking from the opaque box alone, and therefore "obviously" you should take from both. On the other hand, if you do so, then the Being, having foreseen this, would have put nothing into the opaque box, and you end up with only $1,000, rather than $1,000,000; so "obviously" you should forego taking from the transparent box. A truly remarkable quantity of literature has been generated concerning this paradox, much of it densely packed with philosophical jargon and mathematical notations.
After Newcomb had described his paradox to me, I thought it over for a few days and then told him that I would take from only the opaque box. He replied, "Welcome to the Millionaires' Club!" I cannot speak for Newcomb and the other "millionaires", but speaking for myself the only reason I can see for taking from both boxes — which Newcomb called the "greedy" choice — would be because I could not imagine that a Being really could foresee whether or not I would act greedily. However, since I believe that the actions of every person are determined by a definite decision process — that is, by his mental algorithm — it seems that in principle there could be a Being who knows my algorithm and can figure out all its consequences.
A few years after I learned of Newcomb's paradox, while watching the motion picture Hardcore, I noticed a connection between the beliefs of its Calvinist protagonist (played by George C. Scott) and the paradox: If God is omniscient, then he has known since the beginning of time who will be saved and who will be damned. So, if your eternal fate is already established, then why not sin, disbelieve, and/or otherwise offend God if you are so inclined? One answer is that by so doing, you are revealing (in particular to yourself) that you certainly are not among those who will be saved. Similarly in Newcomb's scenario, if you decide to take from both boxes, you are revealing that you are the kind of person whom the Being would have recognized as not deserving of $1,000,000. When I mentioned my observation to Newcomb, he told me that some people have called his paradox "The John Calvin Memorial Paradox".
Newcomb's paradox becomes more complicated if your decision process includes random elements (e.g., coin flipping). If it does, then the description of the paradox needs clarification. Can the Being also predict the outcomes of random events? If not, how does he treat their occurrence? Because these questions did not occur to me while Newcomb was alive, I never asked him about them. One could analyze the situation using the mathematics of probabilistic game theory, but I think that so doing would lead us too far afield from Newcomb's intended point. So let us here assume that the Being withholds the $1,000,000 if he foresees that, instead of making the decision for yourself, you will adopt whatever decision is made by someone or something else (e.g., a flipped coin).
I next shall discuss a connection between Newcomb's paradox and the time travel paradox.
January 18, 2010
